Hölder- and Minkowski-type inequalities for generalized quasi-arithmetic means

Key Points

• The aim is to establish conditions for the validity of functional inequalities involving generalized quasi-arithmetic means.
• Derivation of necessary and sufficient conditions for various inequalities.
• Examination of specific cases involving Hölder, Minkowski, and Jensen inequalities.
• Definition of generalized quasi-arithmetic means using strictly monotone generating functions.
• Established conditions for the validity of several functional inequalities.
• Demonstrated the distinctive properties of generalized quasi-arithmetic means compared to classical definitions.
• Provided insights on how varying the generating functions affects the inequalities.